I'd been meaning to see Touching the Void since its release in early December, but by late January it was down to a daily showing at my local multiplex, a sure sign it was about to disappear. The following week, resigned to waiting for the video, I noticed it was back to three screenings a day. You had to book to be sure of getting in, even so. When I finally saw it, in February, every seat was taken.

The box office success of Touching the Void is hard to credit. It is, after all, a documentary. That was partly why, despite the glowing reviews, I'd dragged my feet - because the word suggests something earnest, pedagogic, propagandist, short on colour and narrative excitement, and more suited to the small screen than the big. Documentary, said its most famous British exponent John Grierson, is about "the creative treatment of actuality". And, according to TS Eliot and Hollywood producers, humankind - or the portion of it that spends Saturday night at the movies - can't bear very much actuality.

Yet the actuality of Touching the Void is its greatest asset. Much of the film consists of interviews with Joe Simpson and Simon Yates about the catastrophe that overtook them in the Peruvian Andes in 1985. To reconstruct that climb, shot on location in Peru and the Alps, the film's director, Kevin Macdonald, uses two actors. But the actors speak even fewer lines than Colin Firth in Girl with a Pearl Earring - all we hear from them are grunts, gasps, screams and fucks.

The film keeps cutting back from the snow to the real-life protagonists as they relive their experience. Rather than breaking the spell, the presence of these men, unfussily filmed (simple headshots, against a plain backcloth), adds to the intensity of the experience and deepens the psychological drama. We know Yates and Simpson survived because they're there, at the campfire, to tell the tale, but that doesn't lessen the suspense.

Beautiful to look at and gripping to watch, Touching the Void enlarges one's sense of what documentary can achieve. Though there's no political or ethnological impulse behind it, its story of isolated human figures battling against the wintry elements is as committed to fact as the first great documentary classic, Robert Flaherty's 1922 Nanook of the North: we hear the clink of crampon and ice axe, see clouds pouring over summits, and are told all about Andean snow, its meringues, mushrooms, cornices and flutings. There's no need for a shaky camcorder. The deal is that this really happened. Any hint of lying or fakery and the film would fall apart.

Macdonald is not alone in extending the documentary tradition. Next month sees the release in Britain of two other compelling examples. In the Oscar-winning The Fog of War, under the gaze of director Errol Morris, the former US defence secretary Robert McNamara looks back over a career that took him and his country into the Vietnam war. And in Andrew Jarecki's Oscar-nominated Capturing the Friedmans we watch the destruction of a family as two of its members, a father and son, stand trial for sexually abusing schoolboys.

These two films are as different from each other as from Touching the Void. But there are important strands in common. All three rely heavily on interviews: as the subjects painfully revisit events from years ago, they narrate to, plead with, explain and exonerate themselves in front of the camera. All three films forgo the assistance of an omniscient narrator, and let the subjects tell their own stories. Most crucial, through a variety of techniques - testimony, photographs, newsreels, 8mm home movie footage - all three constantly remind us these stories are true.

The resurgence of documentary is a phenomenon few could have predicted. The battle for a place in mainstream cinema outlets was lost half a century ago, and in recent years, as reality TV has taken over the schedules, the question has been whether documentary can survive even within mainstream television.

Britain has always been rich in documentary-makers - from Grierson and Humphrey Jennings to Nick Broomfield and Molly Dineen. But since the mid-90s, the space for documentary on television has increasingly been given over to programmes about self- or home-refurbishment - how to improve your cooking skills or your life. So marginalised have documentaries become that the idea of them attracting large cinema audiences, or being borrowed as DVDs and cassettes from video stores, seems wildly implausible.

Yet this has begun to happen. Among prime examples of the revival are Michael Moore's Bowling for Columbine, which extends the territory - idiosyncratic investigative reportage - Moore first occupied with Roger and Me; Spellbound, a documentary about the 1999 US national spelling bee, which draws us into the stories of eight competitors and the very different families and communities to which they belong; Etre et Avoir, which traces a year in the life of a French rural school; and Startup.com, which follows two childhood classmates, Kaleil and Tom, from the launch of their internet venture in May 1999 to its demise on New Year's Day 2001. Mix these in with the biopics (Iris, Sylvia, etc), politico-historical reconstructions (One Day in September) and some outstanding sport and music documentaries (When We Were Kings, The Filth and the Fury), and it begins to look as if more people are watching fact-based movies now than at any point since the second world war.

It would be easy to get overexcited. I can't see The Fog of War or even Capturing the Friedmans enjoying the same success here as they have in the US. And the crossover into biopic and drama-doc (which also includes the part-acted American Splendor) is precisely what directors like Kevin Macdonald, purist about their contract with an audience, seek to avoid. But the hunger for truth - the need to hear what Seamus Heaney calls "the music of what happens" - seems genuine enough. And since documentaries are so cheap to make, there's every reason for producers and distributors to get behind them, so long as the outlet is a multiplex with bums on seats, not the enclave of a film festival or cultural studies department.

The public craving for authenticity explains the predominance of non-narrative fiction - memoirs, life writing, popular history, plays made from the transcripts of court cases - in recent publishing and theatre. Reality TV was another response but it hasn't met the demand, since its reality quota, never large, is now minuscule. Feature films can't satisfy it either. When directors are answerable to committees or an invisible chain of command, the prospects of them achieving authenticity in their movies are diminished.

The budget for Touching the Void was a modest £1.5m. Any US studio optioning Joe Simpson's book (as one has now done, to make a fictionalised version with big-name actors) would spend 10 times as much. But the bigger the budget, the less control for the auteur - and the fainter any semblance of reality.

As the popularity of the Lord of the Rings trilogy shows, the vogue for documentary hardly signals a revulsion against fantasy. But what orcs, wizards and special effects can't accommodate is the need for believable stories and recognisable dilemmas - and for characters whose lives go on after the lights have come up.

"Joe continues to climb to this day," the end of Touching the Void tells us. And, rightly judging that we'll we want to know whether things have turned out OK for the protagonists, the closing credits of Startup.com reveal that Kaleil and Tom have now started a dotcom to help distressed dotcoms.

Fiction sometimes provides a similarly reassuring epilogue - as in "Reader, I married him" (Jane Eyre) or "Viewer, we did it just once but I had his baby" (Cold Mountain) - but in the end, after all the tears, we know novels and feature films are just stories. What documentary-makers do by appropriating the devices of traditional storytelling - especially the use of reliable narrators - is to bring an emotional power to the form that pioneers like Grierson didn't envisage or (seeing it as part of the Hollywood star system) deliberately rejected.

In the days when documentaries were made by behalfers - the socially concerned few speaking on behalf of the silent majority - the mission was to change the world or inspire a sense of nationhood. To his colleague Edgar Anstey, the crime of Humphrey Jennings's wartime classic Listen to Britain was that its elegiac beauty would "not encourage anyone to do anything at all". Now that documentaries (Michael Moore's apart) are no longer politically engagé , pleasing an audience, rather than trying to improve it, seems less of a crime. Even Moore is anarchic and gets his message over with a sense of humour that Grierson, with his Calvinist solemnity, would have distrusted.

Because the stories that documentaries choose to tell aren't determined by focus groups, they're a place for surprising things to happen and for difficult issues to be explored, whether the morality of war, or the rights and wrongs of pushing your kids (Spellbound), or appropriate sexual behaviour.

Jarecki has said of his film about paedophilia, Capturing the Friedmans, that it's designed to provoke rather than lead us to a conclusion: at the end, we're not supposed to think something in particular, "just to think". His method is to lead us through the same twists and turns he must have gone through with the Friedman case, allowing us to assume x, then pulling the rug from our feet with y, then qualifying that with z - and so on until the end of the film, when we're still left puzzling over the guilt of both father and son.

Jarecki succeeds brilliantly, because he had access to tapes and videos made by the Friedmans themselves, a family of manic talkers and inveterate home movie-makers. In the shock of seeing his father on sex charges, the eldest son, David, began filming the family, and went on doing so up to and beyond the trial. The home footage, intercut with local news reports and later interviews, gives a unique sense of drama - we see the family agonising, confessing, denying and playacting, before they know what sentence will be passed. The footage is so intimate that at one point David, in a video diary, warns: "This is private - so if you're not me, you really shouldn't be watching this, turn it off."

Voyeurism is an inescapable part of the modern documentary. Ours is an era of lost inhibitions. There's a pleasure in seeing people exposed to danger or embarrassment: no more shame or stiff upper lips, let the tears flow and the skeletons be brought from the cupboard. Jarecki understands that. Part of the frisson of watching his film is a sense of encroachment. How come we're being allowed in on this stuff? Why is the family (one of the three sons aside) so cooperative? And yet there's nothing exploitative about Jarecki's method. He respects his subjects - and respects his audience to make up its own mind.

This doesn't mean that his film is so wide open as to lack a point of view. Jarecki highlights the untrustworthiness of one witness to the alleged sex abuse by filming him lying on a bed as he makes his accusations, his face in half-shadow and with what looks like an erection beneath his shorts.

Morris's The Fog of War - subtitled Eleven Lessons From the Life of Robert S McNamara - works similarly, balancing admiration for McNamara's willingness to examine his past against condemnation of his complicity in the Vietnam war (and, by extension, Bush's in Iraq). The film begins with two brief sequences, from the 1960s and now, that show McNamara as a media manipulator and control freak, whose sincerity isn't altogether to be trusted.

To bring us exclusives, documentary film-makers, like journalists, depend on a mixture of luck and cunning. Morris had it in finding an octogenarian statesman, with all his intellectual faculties intact, who could be persuaded to be interviewed over 23 hours rather than, as originally agreed, a mere two. The resulting film has earned McNamara a new attention and affection that he couldn't, at 87, have expected.

But it would be surprising if he didn't feel a little traduced by Morris. He comes across in the film as fiercely intelligent and (less predictably) deeply passionate; one section shows his chin trembling and eyes filling as he recalls the the death of JFK. But other shots - of a tough old sonofabitch, with a predator's lower jaw and crooked teeth, cantankerously wagging his finger and stubbornly avoiding awkward questions - are less forgiving.

Every journalist, Janet Malcolm famously said (and her comments equally apply to the documentary film-maker), "is a kind of confidence man, preying on people's vanity, ignorance or loneliness, gaining their trust and betraying them without remorse". This seems too harsh.

Simon Yates is reportedly ambivalent about Touching the Void, and had a difficult time on location in Peru. But the film is scrupulous in avoiding any criticism of him as "the man who cut the rope" and he emerges from it as winningly as Joe Simpson.

"The relationship between director and subject can become very intense," Macdonald says. "It's a bit like therapy, with lots of transferences going on. It's easy to feel guilty."

Jarecki makes a joke of his guilt, and of his subjects' motives in courting publicity, by beginning his film with the song Act Naturally: "They're going to put me in the movies/ They're going to make a big star out of me."

It's reality TV, though, with its 15 minutes of stardom, that's the bigger Judas. A documentary made for the cinema, shot over a lengthy period and based on a degree of mutual trust, needn't diminish its subject. Which is another reason for the appeal of documentaries like Touching the Void and The Fog of War - they don't leave us feeling cheap for having watched them.

Authenticity is a tricky business. The average television documentary isn't big enough to work in the cinema: there's not the depth or originality. Equally, cinema documentaries may have to be edited down for television. Watching a videotape of Touching the Void at home, a fortnight after seeing it at the cinema, I noticed the music more, and the leisureliness, and the mythical subtext beneath the facts, with Joe Simpson as the archetypal wounded hero. It seemed a lusher, more romantic film.

Where it scores, wherever you watch it, is in its narrative power. That's the trick of the current wave of documentaries, even essayistic ones like The Fog of War. They know a good story when they see it (Nick Broomfield's Aileen Wuornos film, for example, is the direct forerunner of Monster). And they've discovered methods of storytelling that encourage us to think and feel for ourselves, rather than feeding us the party line.

"If documentary is going to be significant," Paul Rothe wrote in 1935, "we must make films which will move the people and not just amuse our fellow directors." Take the word "the" out and that could stand as a manifesto for current practice.

· The Fog of War is released on April 2, and Capturing the Friedmans on April 9.